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Maths and Sport

A question of balance

These project ideas explore moment of inertia and its consequences in a number of different sports, and encourage investigation, experiment and discussion. This is aimed at higher level GCSE and A Level students (Key Stages 4 and 5).

A question of balance

What factors will improve your balance on a gymnastic beam or tight-rope? Why do tight-rope walkers often carry long poles? Try standing with your feet placed one behind the other. Why is this more difficult than standing normally?

When balancing stability is increased by:

being symmetrical about the centre

distributing mass far away from the centre

lowering the centre of gravity

widening the base of support

The mathematical concept involved here is that of inertia. An object which has high inertia is slow to move, an object which has low inertia is easy to move. Mathematically, inertia is related to the mass of the object, its average radius, and the way that mass is concentrated relative to the centre. In this equation, the constant C shows how much the mass is concentrated towards the centre of the object. A high value indicates that mass is concentrated towards the outside of the object, a low value that it is concentrated towards the centre of the object, as can be seen in the diagram on the left. The value of C for the hollow shell is 2/3, whereas for a solid sphere of the same mass (different density), it is only 2/5.

Sports where inertia is a factor to consider include skating, diving, gymnastics and cycling. When skaters want to move more rapidly, they put their arms close to their bodies (as in the diagram on the left), when they want to move more slowly, they stretch their arms out (as in the diagram on the right. You can find a nice animation illustrating skating jumps and turns on the Olympic.org website).

Ordinary cycle wheels look like those in the left-hand diagram of the pair on the right, racing cycle wheels look like the right-hand diagram. In an ordinary cycle wheel, the mass is concentrated towards the outside, as with the hollow shell. This increases the value of C, and thus the inertia, helping us to maintain our balance. The spokes mean that there is turbulence in the air around the wheel, however, and to reduce air resistance and improve aerodynamics racing cyclists therefore prefer solid wheels, like that on the right. How does the inertia of a solid wheel compare to an ordinary wheel? The project below asks you to think about how it could help a racing cycle move more quickly.

Projects

Inertia, like mass and weight, is an important factor to consider in a number of different sports. Mass is determined by the number of elementary particles of matter that make up the body. Its weight is determined by the gravitational force on that mass in the Earth's gravity field and is given by the product of the mass and the acceleration due to gravity, W = mg.

The mass and the weight of an object do not tell us everything we need to know when we are considering sports: they do not tell us how hard it is to move or rotate an object. This depends on the distribution of the mass. Find some objects and see which are easier to move and rotate.

You should find that if all the mass is concentrated near the centre then an object will be easier to move than one of the same diameter and mass (made of material of different density) in which all the mass is near the edge. A hollow shell is harder to roll than a solid ball. This property of objects is captured by defining what mathematicians call the moment of inertia, I. It is given by the mass multiplied by the square of a radius of mass concentration (called the 'radius of gyration'). For example, a solid sphere of mass M and radius R has

but a hollow spherical shell of mass M and radius R has

If you disturb an object with moment of inertia I then typically the time to make it move a given amount is proportional to the square root of I so the larger I is, the more time it takes to move the object, while the smaller the value of I, the easier it is to move it. In some situations, like balancing on the tight-rope you want to make the time it takes you to wobble as long as possible so that you can counteract it easily.

Why do racing cycles have solid wheels rather than hollow wheels with spokes linking the hub to a solid exterior wheel?

Why do you think that weight-lifting barbells have weights attached to the ends of a bar?

Can you think of other Olympic events where the equipment used is influenced by its moment of inertia?

In high-board diving and gymnastics the athletes often perform somersaults. How does their moment of inertia affect their ability to complete somersaults in mid-air? What do they do to change their moment of inertia?

Background information

The Olympic.org website includes animations for each sport illustrating techniques and equipment used.

The BBC Sport website has a guide to track cycling illustrating the differences between a track bike and an ordinary road bike.

These project ideas were originally developed by our Motivate project as follow-up activities to a talk by Professor John D Barrow FRS on 'The Maths Behind the Olympic Games'.